Article pubs.acs.org/JPCA
More Protected Vibrational States at the Dissociation Limit of SCCl2 Eduardo Berrios,† Susan Pratt,‡ Prabhat Tripathi,† and Martin Gruebele*,†,‡,§ †
Department of Chemistry, ‡Department of Physics, and §Center for Biophysics and Computational Biology, University of Illinois, Urbana, Illinois 61801, United States S Supporting Information *
ABSTRACT: Local vibrational coupling models predict that intramolecular vibrational energy redistribution (IVR) is not completely statistical even at the dissociation limit of polyatomic molecules. Thus states protected from IVR and from rapid dissociation form regular progressions and can be assigned vibrational quantum numbers. We previously observed such regular progressions of states in vibrational spectra of the molecule SCCl2, but a discrepancy in the density of such states remained between theory and experiment. Here we show that the gap can be closed by observing and assigning additional vibrational transitions above the dissociation limit of SCCl2, and by carefully analyzing the theoretically expected density of protected states. The newly observed transitions originate from recently assigned and more highly excited vibrational levels in the B̃ electronic state, connecting to the X̃ ground state by different Franck−Condon factors. Based on our analysis of Franck−Condon activity, we conclude that theory and experiment agree within measurement uncertainty. Consistency between theory and experiment implies that even more protected states should be observed for larger molecules, leading to nonstatistical dissociation reactions.
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tiers17). Such models often rely on simplified potential energy surfaces to treat the molecule at a level between simple statistical models and full ab initio calculations.18,19 In analogy to periodic orbits embedded in an otherwise chaotic classical phase space, some states escape the network of anharmonic vibrational couplings that allows energy to flow.20,21 Local coupling models have predicted phenomena from fractal wave functions,22 to power law energy flow,23 to the existence of surprisingly large numbers of regular progressions and assignable states at the dissociation energy of sufficiently large molecules.24 Signatures of these phenomena have been observed experimentally,6,9,13,25 and agreement between local coupling model predictions and experimental observations is often qualitatively good.26,27 The question then is: are the discrepancies just due to experimental or computational limitations, or are there still fundamental ingredients missing in the models? A case in point is the vibrational states of SCCl 2 (thiophosgene) near dissociation. If energy flowed freely, one would expect all observed bright states to have statistically distributed energies, and to be fragmented into clumps of eigenstates because of coupling with nearby dark states. Figure 1A schematically illustrates such a distribution of bright states: very few bright states are undiluted (dilution factor σ = 1 means
INTRODUCTION Vibrational energy should flow freely in molecules at the dissociation limit. At least that is the assumption of simple statistical reaction rate models. Unless one gets very close to the reaction threshold,1 experiments on small molecules like NO2 support this notion,2−4 yielding near-dissociation spectra whose randomly distributed line positions and widths can be modeled by global random matrix models.5 Classically, such behavior corresponds to chaotic dynamics. However, spectroscopic studies of other small molecules have also revealed some order near the dissociation limit. For example, Choi and Moore6 have observed regular progressions of vibrational states near the dissociation limit of OCFH by using stimulated emission pumping (SEP).7 Vibrational spectra of water above its dissociation limit have been recorded9 by action spectroscopy8 up to 19 quanta of combined excitation, and ab initio analysis is now able to reproduce such spectral features.10 Studies of HOCl O−H stretching overtones show that energy does not flow statistically into the reaction coordinate near the dissociation limit.11 Regular polyad structures persist in acetylene molecules at energies that allow isomerization to vinylidene.12 Laser-induced fluorescence and SEP have revealed regular progressions of vibrational states with up to 35 combined quanta above the dissociation limit of SCCl2,13 where the total anharmonic density of vibrational states exceeds 5 × 106/THz. Local coupling models attempt to explain the orderly spectra of molecules excited to “chemical” energies.14−16 These models take into account that the matrices describing anharmonically coupled vibrational states can be highly structured (e.g., into © XXXX American Chemical Society
Special Issue: Curt Wittig Festschrift Received: May 16, 2013 Revised: September 12, 2013
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with theory if Franck−Condon activity is taken into account. We provide two lines of evidence that the local coupling model based on a simple resonance Hamiltonian and the experimental data are actually in quite good agreement. First, we measure many additional transitions above the dissociation limit of SCCl2, particularly those active in ν2 (C−Cl symmetric stretching mode) and ν3 (Cl−C−Cl symmetric bending mode) that have low Franck−Condon activity, nearly doubling the experimentally observed density of protected states in the 600 to 620 THz range to ≈1.9 states/THz. This new search for additional protected states was made possible by a better assignment of the vibrational character of the SCCl2 B̃ state, from which fluorescence to the X̃ state dissociation limit is observed.35 Second, we analyze the states from the resonance Hamiltonian model more systematically, decimating the full set of vibrational states according to symmetry, according to a dilution factor pattern mimicking experiment, and according to Franck−Condon activity similar to that seen in the newest experiments. When thus filtered to quantitatively mimic the experimental limitations, the resonance Hamiltonian model predicts ≈1.9 observed protected states/THz in the 600 to 620 THz range. Local coupling models of energy flow predict that the fraction of protected quantum states increases with molecular size:24 The excitation energy for breaking a chemical bond is always about the same for any molecule, but in large, irregularly shaped molecules energy flow becomes arrested once the energy is spread “too thin” over many partly localized vibrational modes. Thus SCCl2 may be just the tip of the iceberg. Indeed, large deviations from statistical rate models have been observed in much larger molecules,37,38 and explained using local coupling models.39,40 The local coupling theory and experiments consistent with it bode well for finding regular progressions in much larger molecules than SCCl2, such as amino acids in He droplets.
Figure 1. (A) Global anharmonic mixing. Eigenstates are shown as sticks, resonances above dissociation as Lorentzians. The dilution factor σ characterizes into how many eigenstates or resonances a bright state is mixed by IVR (e.g., the bright state with σ ≈ 1/10 is mixed into 10 eigenstates). When mixing is global, most dilution factors σ are small and line spacings are irregular (Brody-distributed36). (B) Only a local density of states contributes to mixing.16 This corresponds to the case in classical mechanics where regular motion coexists with chaotic motion. Some bright states are “protected” from IVR and correspond to a single eigenstate or resonance, while others have small dilution factors. The protected states form regular progressions (dashed curves) that can be assigned vibrational quantum numbers, although these quantum numbers do not necessarily correspond to simple normal modes.
a bright state corresponds to a single estate), and many bright states are diluted (σ = 1/10 means the bright state is mixed over 10 eigenstates).28 Instead, LIF and SEP experiments revealed that SCCl2 exhibits several completely regular vibrational progressions (illustrated in color in Figure 1B) up to at least 90 THz (∼2700 cm−1) above its first dissociation limit (at. ∼597 THz or ∼20,000 cm−1).13,29 The progressions can be classified by polyad quantum numbers.30−32 We say that such states are protected from IVR and thus call them “protected states”. A computational analysis by Chowdary and Gruebele,33 based on a resonance Hamiltonian of Sibert and Gruebele,34 indicated there should be roughly 10 times more protected states than were observed experimentally.29 The difference was ascribed to two causes: (1) only A1 symmetry states can be observed, but states with other symmetries can also be protected; (2) insufficient Franck−Condon activity of symmetry-allowed transitions further cuts the density of protected states observed in ref 29 to only ≈1.1 states/THz [In ref 29, this is misprinted as 0.005/GHz at 660 THz, whereas a count from Table 1 in ref 29 gives 0.0005/GHz or 0.5/THz. From 600 to 620 THz, the region we use here for comparison because of the higher signal-to-noise ratio and hence better statistics, the value from Table 1 in ref 29 is 1.1/ THz. The total density of states quoted in ref 29 is a harmonic count. Here we use the anharmonic count for a more quantitative comparison.] in the 600 to 620 THz range just above the dissociation limit. However, a number of assumptions were made in the computational analysis that made it difficult to account for the difference quantitatively. We show here that the total number of protected states should exceed the observed number by far more than a factor of 10 based on the anharmonic state density, but that the number that is expected to be seen by experiment matches up very well
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METHODS Experimental Setup. Our laboratory setup to measure laser induced fluorescence of jet-cooled thiophosgene has been discussed in detail elsewhere.13 Thiophosgene seeded into helium at approximately 5% was expanded through the 500 μm diameter orifice of a piezoelectric valve operated at 20 Hz and 105 Pa backing pressure. The resulting jet expanded into a vacuum chamber held at 0.01 Pa by a diffusion pump. The internally cold SCCl2 molecules (∼10 K rotational temperature, 100 K vibrational temperature) met the excitation laser pulse at a right angle about 5 mm from the nozzle. Fluorescence was collected perpendicular to the molecular beam and the laser pulse, and steered to a 0.75 m monochromator where it was dispersed through a slit onto a photomultiplier tube. When dispersed fluorescence near the X̃ state dissociation limit was collected, a long-pass filter was placed before the monochromator to suppress second-order diffraction bleed-through. The photomultiplier signal was monitored by a gated boxcar integrator averaging over 200−400 shots per point and digitized to a personal computer for data analysis. The excitation laser pulse was obtained by pumping a dye laser (ND6000, Continuum) with the 532 nm second harmonic from a Nd:YAG laser (Surelite II, Continuum). Rhodamine 590 or Fluorescein 548 was used as the gain medium, depending on the initial vibrational state to be addressed in the B̃ electronic manifold. The dye laser output was frequency doubled by an B
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where the first term is a polynomial expansion as in eq 1, and the second term accounts for anharmonic resonances via unitless raising operators ar and lowering operators ar. Specifically, we included only the four most prominent vibrational resonances classified by Sibert and Gruebele33,34
Autotracker III (INRAD) to ensure the pulse intensity was constant to ±5%. Vibrational Hamiltonian. We used two different vibrational Hamiltonians for two different purposes. The line centers of newly assigned vibrational transitions near dissociation, together with all previously assigned X̃ state transitions13,29,41,42 were fitted to an effective Hamiltonian with no explicit resonances: 6
Evib /h =
∑ vr ⎛⎝nr + ⎜
r=1
1 ⎞⎟ + 2⎠
6
∑ r = 1, s > r
Hres (in units of GHz) = −323a 2†a5a6† − 306a1a5†a6† + 121a1†a 2†a52 − 24a32a62
⎛ 1 ⎞⎛ 1⎞ χrs ⎜nr + ⎟⎜ns + ⎟ ⎝ ⎠ ⎝ 2 2⎠
where “c.c” indicates addition of the Hermitian conjugate. Eigenstates of Hdiag in eq 2 are the zero order basis set. To calculate the dilution factor σ for a specific bright state |i⟩ = |ni1...ni6⟩, a truncated basis set was constructed including the bright state, all zero order states coupled to it directly by eq 3, and additional tiers of zero order states coupled indirectly via other zero order states. To obtain a manageable basis set size, the following restrictions were applied: only states with the right symmetry are included; only states inside a 60 THz energy window are included; only 5 tiers were included after the bright state; a cutoff distance of ∑r=1...6|nir − ni′r|≤7 between bright states |i⟩ and |i′⟩ in state space was enforced. Matrix diagonalization yields approximate vibrational eigenstates |j⟩ whose energies Ej and projections |⟨i|j⟩|2 = Ii (Ej) = Iij yield the spectral envelope into which the bright state has been diluted. The dilution factor of the bright state |i⟩ is calculated as (see Figure 1):
⎛ 1 ⎞⎛ 1 ⎞⎛ 1⎞ + ∑ χrst ⎜nr + ⎟⎜ns + ⎟⎜nt + ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ 2 2 2⎠ r = 1, s > r , > s (1)
This is the same effective Hamiltonian that was used previousy to fit vibrational progressions in the 6 vibrational modes of SCCl2.13,29 Modes 1 through 6 are summarized in Table 1. Table 1. Effective Hamiltonian Parameters of Equation 1 for the Fit of All Assigned States from the Origin up to 690 THza fitted parameters
value
2σ
34.602 15.427 8.971 14.274 24.641 8.979 −0.133 0.001 0.017 −0.113 −0.228 −0.090 −0.050 −0.190 −0.069 −0.034 −0.015 −0.006 −0.060 0.405 −0.0016 0.0005 −0.0805 −0.0016 0.0283
0.015 0.078 0.051 0.008 0.078 0.448 0.001 0.006 0.004 0.001 0.021 0.004 0.018 0.062 0.006 0.011 0.004 0.001 0.004 0.152 0.00007 0.00006 0.0057 0.0006 0.0066
σi =
∑j Iij2 (∑j Iij)2
0 < σi ≤ 1 (4)
Going back to the example in the Introduction, if the spectral intensity of the bright state i had been evenly diluted over 10 eigenstates j, each with intensity Iij = 1/10, then σ = 10·(1/ 10)2/(10·1/10)2 = 1/10. The Hamiltonian in eq 2 was fitted to data only up to ca. 300 THz, and underestimates slightly the diagonal anharmonicity. Compared to eq 1 fitted to experimental data, eq 2 predicts energies too high by about 30 THz near the dissociation energy, so when comparing computed statistics of σ with experiments, simulated states in a correspondingly shifted energy window will be used.
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RESULTS
Selection of Upper States for Emission. Vibrational modes of SCCl2 are identified in Table 1. Chowdary and Gruebele29 previously reported emission from the ν1 and ν1+ν4 vibrational levels of the B̃ state to near the dissociation limit of the X̃ state. In a recent reanalysis of the B̃ state, it was shown that the ν1+ν4 state has significant ν2+ν4 character.35 We picked a number of higher-lying vibrational levels in the B̃ state based on the reanalysis, hoping that their Franck−Condon activity would be different from the previously assigned spectra. Figure 2 shows an overview of the fluorescence experiment, and excitation spectra around three new upper states: 2ν1, 3ν2, and 3ν2+ν4 (these are nominal assignments; Fermi resonance of ν2 and 2ν3, as well as fourth order resonance of ν1 and ν2 mixes the character of those states). Figure 2 also shows a lowresolution overview of the emission spectra thus obtained, which exhibit very different Franck−Condon activity from previous data, in particular the 3ν2+ν4 transition at the highest
a
Frequencies are identified by mode symmetry and qualitative character. Parameters values and 2 standard deviation uncertainties are in THz.
Such a diagonal Hamiltonian of course cannot be used to compute dilution factors because it assumes that all bright states correspond to a single vibrational eigenstate. We calculated dilution factors for vibrational states near the dissociation limit by using the resonance Hamiltonian developed by Sibert and Gruebele.34 The resonance Hamiltonian has the form H = Hdiag(nr ) + Hres(ar , ar )
(3)
+ c.c.
6
v1 (a1 CS stretch) v2 (a1 CCl sretch) v3 (a1 CCl bend) v4 (b2 umbrella) v5 (b1 CCl stretch) v6 (b1 CCl bend) χ11 χ12 χ13 χ14 χ15 χ16 χ22 χ23 χ24 χ33 χ34 χ44 χ46 χ66 χ114 χ144 χ123 χ124 χ234
†
(2) C
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Figure 2. Laser induced fluorescence is collected from vibrationally excited states in the B̃ state of thiophosgene. Sample excitation spectra are shown for bands that lie 29.73 THz, 43.48 THz, and 52.41 THz above the band origin of the B̃ state at 1027.598 THz (black dots). The bottom right panel displays dispersed fluorescence to the X̃ state collected from these three bands in two energy ranges: 0 to 300 THz and 550 to 680 THz.
excitation energy. In addition, the 2ν2+ν4 band and the ν2+3ν4 band were also pumped and their emission observed. New Regular Progressions of Vibrational States. Figure 3 shows two of the emission spectra expanded around the first dissociation energy of SCCl2 (∼597 THz to CS+Cl2; the second lies at ∼666 THz to S + SCCl). In panel A, dispersed fluorescence from the 3v2 upper state is plotted from 550 THz up to 680 THz. Regular progressions can be assigned to combinations of v1 and v4, with added excitation in the ν2 mode. Consecutive clusters marked by red lines in Figure 3 are obtained by adding one quantum of v1 excitation. The first two clusters were previously observed as weak progressions upon excitation to v1 in the B̃ state.29 Here, new progressions are observed at energies higher than the SCCl2 molecular dissociation channel at 597 THz. These new progressions are listed in Table 2 for the 600 to 620 THz range, together with all new and old assigned and unassigned “sharp” ( 0.5 calculations, again showing a Franck−Condon bias in the experimental observations.
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DISCUSSION Just because the vibrational spectrum of SCCl2 can be fitted by a single expression (eq 1) from zero point to dissociation, does not guarantee that the vibrational wave functions at high energy are just mildly distorted by anharmonicity. Our reduceddimensional computational analysis in ref 29 showed that states below 350 THz are harmonic-oscillator like, whereas states above 500 THz had locked the ν1 (CS stretch) and ν4 (umbrella) vibrations together in phase, producing arc-shaped wave functions in a plot of normal coordinates q1 vs q4. The classical interpretation is that as the CS bond stretches (ν1), the G
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density of states, in good agreement with an analytical model for the fraction of protected states24 as evaluated in Figure 4 of ref 33. Finally, we estimated how many of these states would be missed because of insufficient Franck−Condon activity in the experiment: in the experimental data from 600 to 620 THz, only 8 < n1 < 18, n2 < 4, n3 < 4, 7 < n4 < 23 and n5 = n6 = 0 were observed. Assuming statistical independence of the quantum numbers (valid over a sufficiently small energy range), and applying the same restriction to the 1 × 105 computed protected states with σ > 0.5 and A1 symmetry, yielded 38 states (sideways triangle in Figure 7). We observed 37 experimentally (black triangle in Figure 7). For energies with Poisson-distributed spacings, one expects a variation of about ±3 states over a 20 THz window with a state density of 37/20 THz ≈ 1.9/THz, so this agreement is within the expected statistical range. Reasonable variations of this scheme (e.g., using the distribution of 0.3 ≤ σobs ≤ 0.9 from Table 2 instead of σ > 0.5) do not alter the result. Therefore there is no significant discrepancy between experiment and the resonance Hamiltonian model, and the resonance Hamiltonian model33 also agrees very well with the analytical model for the fraction of protected states among all vibrational states.24 In summary, we have narrowed the gap between the number of experimentally observed and theoretically predicted protected states at the dissociation limit of SCCl2. We provide convincing evidence that the missing protected states can be explained by symmetry and insufficient Franck−Condon activity especially in the ν2 (CCl stretching) and ν3 (CCl bending) modes from accessible B̃ state vibrational levels. To the extent that this agreement validates local coupling models, we expect that such protected states will be even more numerous for larger molecules because the average energy per mode will be even smaller in large molecules upon energy redistribution, leading to weaker coupling. Hopefully such states can be observed in large molecules by the new generation of ultracold spectroscopy in He clusters.
shifts from 330 THz to 317 THz: its vibrational wave function attains the arc shape characteristic of stretch−bend locking. So even the weak q1−q4 CS stretch-umbrella bend coupling begins to switch the character of the wave functions from nearly separable to highly correlated above 300 THz. The remixing is complete above 500 THz, and the “phoenix emerges from the ashes” in the form of regularly spaced vibrational progressions that extend above both CS+Cl2 and C +SCCl dissociation channels. The remaining question is whether the local coupling model prediction33 is truly off from the observed density of bright states. Table 2 shows that 15 transitions have been added to the previous 22 in the region from 600 to 620 THz,29 by observing emission from five new upper states vs the original two. The signal-to-noise-ratio of both data sets is very similar, so this increase should be representative of the absolute increase obtained by exploring a broader range of Franck−Condon activity. Figure 7 outlines how we compared the computed and observed number of protected states in the interval from 600 to
Figure 7. Computed number of states in ±10 THz intervals, for comparison with experiment at 610 ± 10 THz. The anharmonic state counts based on the diagonal constants from ref 34 and Table 1 are in good agreement, but considerably higher than the harmonic count from ref 29 or the very similar Whitten−Rabinovich45 estimate. The experimentally observed count of IVR-protected states reported here is much lower. When the computed count is filtered to include only states with similar dilution as predicted for experimentally observed states (σ ≥ 0.56), only the experimentally observed A1 symmetry, and finally the Franck−Condon cutoffs observed in the experimental data are enforced, good agreement between the experiment and theory is obtained.
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ASSOCIATED CONTENT
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AUTHOR INFORMATION
S Supporting Information *
The PDF Supporting Information file contains a table of all observed vibrational energy levels of SCCl2, with assignments when made. This material is available free of charge via the Internet at http://pubs.acs.org.
Corresponding Author
*E-mail:
[email protected]. Phone: (217) 333-1624.
620 THz more quantitatively than was done in ref 29. First, we computed the total anharmonic density of states by direct count. We used both the diagonal terms of the Sibert and Gruebele resonance Hamiltonian (red squares), as well as the parameters in Table 1 (black circles). They are in good agreement, with about 108 total states in the 610 ± 10 THz interval, or about 5 × 106/THz. For reference, Figure 7 also plots the harmonic density of states listed in the introduction of ref 29, which agrees well with the Whitten−Rabinovich formula.45 Next, we eliminated all states except those with σ > 0.5, the average value computed for the experimentally observed protected states. This left 4 × 105 protected states, or a fraction of 0.004 of the total anharmonic density of states. Next, we eliminated all states without A1 symmetry because only A1 symmetry states can be observed by LIF. This left 1 × 105 protected states with A1 symmetry, or 0.001 of the total
Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS M.G. was supported by a James R. Eiszner Professorship, and E.B. by a Fulbright-CONICYT Fellowship. The project was funded by an NSF Grant CHE-1012075.
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